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Normal Forms for Semilinear Quantum Harmonic Oscillators

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Abstract

We consider the semilinear harmonic oscillator

$$i\psi_t=(-\Delta +{|x|}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \mathbb{R}^{d},\, t\in \mathbb{R},$$

where M is a Hermite multiplier and g a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on M related to the non resonance of the linear part, this normal form is integrable when d = 1 and gives rise to simple (in particular bounded) dynamics when d ≥ 2. As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions.

Résumé (Formes normales de Birkhoff pour l’oscillateur harmonique quantique non linéaire)

Dans cet article nous considérons l’oscillateur harmonique semi-linéaire:

$$i\psi_t=(-\Delta +x^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \mathbb{R}^{d},\, t\in \mathbb{R},$$

M est un multiplicateur de Hermite et g est une fonction régulière globalement d’ordre au moins trois. Nous montrons qu’une telle équation admet, au voisinage de zéro, une forme normale de Birkhoff à n’importe quel ordre et que, sous des hypothèses génériques sur M liées à la non résonance de la partie linéaire, cette forme normale est complètement intégrable si d = 1 et donne lieu à une dynamique simple (et en particulier bornée) pour d ≥ 2. Ce résultat nous permet de démontrer l’existence presque globale et de contrôler les normes de Sobolev d’indice grand des solutions de l’équation non linéaire ci-dessus avec donnée initiale petite.

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Correspondence to Benoît Grébert.

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Communicated by G. Gallavotti

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Grébert, B., Imekraz, R. & Paturel, É. Normal Forms for Semilinear Quantum Harmonic Oscillators. Commun. Math. Phys. 291, 763–798 (2009). https://doi.org/10.1007/s00220-009-0800-x

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